I recently purchased an AirPort Express and tried it out with the four possible settings under Wireless in the AirPort Utility.
The AirPort Utility was run on a MacBook Air connected to the home WLAN. The home has a LAN/WLAN connected to the Internet by a modemrouterserver furnished by an ISP.
A global comment is that the AirPort Express and the AirPort Utility often failed to work properly after settings were changed. It sometimes helped to reboot both the AirPort Express and the MacBook on which the AirPort Utility was run. (Yes, reboot the machine, not just restart the AirPort Utility.)
* Wireless = Create new wireless network
* Internet = DHCP
* Network = DHCP and NAT
In this mode the AirPort Express connected to the LAN via the WAN port; it received a LAN address via DHCP. One device was connected to the LAN port and received from the AirPort Express the address 10.0.1.2; another connected wirelessly and received address 10.0.1.3. Both devices had access to the Internet.
* Wireless = Extend a wirless network
* Internet = DHCP
* Network = Off (Bridge Mode) [greyed out]
In this mode the AirPort express didn't connect to the home WLAN. Nothing worked.
* Wireless = Join a wireless network
* Internet = DHCP
* [No Network tab is shown]
The AirPort Express joined the home WLAN and obtained an address via DHCP. A device connected to the AirPort's LAN port was able to join the same home LAN via DHCP. Initially a device connected to the WAN port was unable to connect. However, after rebooting both the AirPort Express and the MacBook running AirPort Utility, the device connected to the WAN port was able to join the home LAN via DHCP.
* Wireless = Off
* Internet = DHCP
* Network = Off (Bridge Mode)
The AirPort Express joined the home LAN and obtained an address via DHCP. A client connected to the LAN port was able to join the same home LAN via DHCP.
This is the mode I use. I purchased the AirPort Express in order to be able to put an existing Ethernet-connected Internet radio player (with aux input) to use as an AirPlay renderer.